In order to study the relaxed problem, we rely on duality techniques that originated in the theory of Optimal Transport (see [1]}, [2]}, [3]} and [4]}) and were further developed in the theory of Mean Field Games. Indeed, when the game has a potential structure — see for instance Lasry, Lions [5]}, Cardaliaguet, Graber, Porretta and Tonon [6]}, Briani and Cardaliaguet [7]} and Orrieri, Porretta and Savaré [8]} — the system of partial differential equations which describes the distribution of the players and the value function of a typical infinitesimal player can be obtained as optimality conditions for an optimal control problem for the Fokker-Planck equation. In this framework, the necessary conditions are obtained through convex duality techniques, using generally the Fenchel-Rockafellar theorem as in [6]}, [7]} or the Von-Neumann theorem as in [8]}. We follow this path and — when the final constraint is linear — we are able to exhibit a dual problem, which is an optimal control problem for the HJB equation involving the Lagrange multiplier \(\lambda \in \mathbb {R}^+\)  associated to the terminal constraint. It takes the following form :
\(\sup _{(\lambda ,\phi )} \int _{\mathbb {R}^d} \phi (0,x) m_0(dx)\) 
